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Control-based continuation (CBC) is a general and systematic method toexplore the dynamic response of a physical system and perform bifurcationanalysis directly during experimental tests. Although CBC has been successfullydemonstrated on a wide range of systems, rigorous and general approaches todesigning a noninvasive controller underpinning the methodology are stilllacking. In this paper, a noninvasive adaptive control strategy for a wideclass of nonlinear systems with unknown parameters is proposed. We prove thatthe proposed adaptive control methodology is such that the states of thedynamical system track a reference signal in a noninvasive manner if and onlyif the reference is an open-loop response of the system. Compared to theexisting literature, the proposed method does not require any a prioriknowledge of some system parameters, does not require a persistent excitation,and is not restricted to linearly-stable systems, facilitating the applicationof CBC to a much larger class of systems than before. Rigorous mathematicalanalyses are provided, and the proposed control method is numericallydemonstrated on a range of single- and multi-degree-of-freedom nonlinearsystems, including a Duffing oscillator with multiple static equilibria. It isshown that both the stable and unstable periodic orbits of the open-loop systemcan be stabilized and reached, noninvasively, in closed-loop conditions.

Noninvasive Adaptive Control of a Class of Nonlinear Systems With Unknown Parameters